Information-Theoretic Limits for the Matrix Tensor Product

TL;DR

This paper derives exact formulas for the mutual information and MMSE in high-dimensional matrix tensor product models, generalizing key data science problems like PCA and network analysis, and introduces new analytical techniques.

Contribution

It provides the first single-letter formulas for mutual information and MMSE in high-dimensional matrix tensor models, with novel analysis methods for matrix-valued signals.

Findings

Exact formulas for mutual information and MMSE in high-dimensional regimes.

Non-asymptotic bounds matching the formulas in the leading order.

New analytical techniques for high-dimensional matrix signal analysis.

Abstract

This paper studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse principal component analysis and covariance estimation and the stochastic block model used in network analysis. The main results are single-letter formulas (i.e., analytical expressions that can be approximated numerically) for the mutual information and the minimum mean-squared error (MMSE) in the Bayes optimal setting where the distributions of all random quantities are known. We provide non-asymptotic bounds and show that our formulas describe exactly the leading order terms in the mutual information and MMSE in the high-dimensional regime where the number of rows $n$ and number of columns $d$ scale with $d = O(n^\alpha)$ for some $\alpha < 1/20$. On the technical side, this paper introduces some new techniques for the analysis of high-dimensional matrix-valued signals. Specific contributions include a novel extension of the adaptive interpolation method that uses order-preserving positive semidefinite interpolation paths, and a variance inequality between the overlap and the free energy that is based on continuous-time I-MMSE relations.